Solve left(4-yright)^2+left(2y+1right)^2=5y^2+y

$${ \left(4-y \right) }^{ 2 } + { \left(2y+1 \right) }^{ 2 } = 5 { y }^{ 2 } +y$$

Solve for y

$$y = \frac{17}{5} = 3\frac{2}{5} = 3.4$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-y\right)^{2}.

$$16-8y+y^{2}+\left(2y+1\right)^{2}=5y^{2}+y$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2y+1\right)^{2}.

$$16-8y+y^{2}+4y^{2}+4y+1=5y^{2}+y$$

Combine y^{2} and 4y^{2} to get 5y^{2}.

$$16-8y+5y^{2}+4y+1=5y^{2}+y$$

Combine -8y and 4y to get -4y.

$$16-4y+5y^{2}+1=5y^{2}+y$$

Add 16 and 1 to get 17.

$$17-4y+5y^{2}=5y^{2}+y$$

Subtract 5y^{2} from both sides.

$$17-4y+5y^{2}-5y^{2}=y$$

Combine 5y^{2} and -5y^{2} to get 0.

$$17-4y=y$$

Subtract y from both sides.

$$17-4y-y=0$$

Combine -4y and -y to get -5y.

$$17-5y=0$$

Subtract 17 from both sides. Anything subtracted from zero gives its negation.

$$-5y=-17$$

Divide both sides by -5.

$$y=\frac{-17}{-5}$$

Fraction \frac{-17}{-5} can be simplified to \frac{17}{5} by removing the negative sign from both the numerator and the denominator.

$$y=\frac{17}{5}$$